unQVNTS (Vermont)

Anna Haensch, Duquesne University
An Introduction to Lattices and the Representation Problem

Given a polynomial f(x) of several variables with rational coefficients and an integer n, we say that f represents n if the equation f(x)=n is solvable in the integers. One might ask, is it possible to effectively determine the set of integers represented by f? This so-called representation problem for quadratic polynomials is one of the classical problems in number theory. The negative answer to Hilbert's 10th problem tells us that in general, there is no finite algorithm to decide whether a solution exists. In this expository talk, we will view the polynomial and its solutions as an algebraic construction called a lattice, and we will discuss some important classical solutions to the representation problem, including famous theorems of Euclid, Legendre, Gauss, and Fermat.

Anna Haensch, Duquesne University
Almost Universal Ternary Sums of Polygonal Numbers

In 1796 Gauss showed that every natural number can be written as the sum of three triangular numbers. In 2009, Chan and Oh determined when a weighted sum of triangular numbers with coefficients represents all but finitely many natural numbers; we call such a sum is almost universal. In this talk we will determine when a sum of three generalized m-gonal numbers is almost universal. We will approach this question first from an algebraic, and then from analytic point of view, exploiting the capabilities of each method, and realizing new connections between the spinor exceptions of a lattice and the decomposition of its theta series.

Taylor Dupuy, University of Vermont
Frobenius Angle Rank and Bizzaro Hodge Numbers

A famous theorem of Ax-Katz-Chevalley-Warning give congruences for the number of FF_q points on varieties over finite fields modulo a certain power of q. The power of q in this congruence is related to the concentration of the Hodge diamond of a variety and is called the (normalized) Hodge co-level. In Alexander Mueller's thesis he discovered congruence phenomena holds for certain *non-integral* or "bizarro" powers of q. This phenomena is related to relations among Frobenius eigenvalues. We will talk about what we can say about relations among Frobenius eigenvalues of abelian varieties. This is ongoing joint work with David Zureick-Brown;  Kiran Kedlaya, David Roe and Christelle Vincent.

Christelle Vincent, University of Vermont
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George Melvin, Middleburry College
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